Abstract
The algorithms for stabilizing particular systems of differential equations with a constant delay are proposed; here, the right side of one of subsystems contains the multiplier e t. The sufficient conditions of the instability and the stability in a first approximation are obtained.
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References
E. A. Barbashin, Introduction to Stability Theory (Nauka, Moscow, 1967) [in Russian].
L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Nauka, Moscow, 1971; Academic, New York, 1973).
S. N. Shimanov, “Instability of motion of a system with delay,” Prikl. Mat. Mekh. 24, 447–457 (1960).
G. M. Fikhtengol’ts, Course of Differential and Integral Calculus (Fizmatlit, Moscow, 2002), Vol. 3 [in Russian].
Yu. A. Mitropol’skii, Averaging Method in Nonlinear Mechanics (Vishcha Shkola, Kiev, 1971) [in Russian].
B. G. Grebenshchikov, “The first-approximation stability of a nonstationary system with delay,” Russ. Math. 56, 29–36 (2013).
R. Bellman and K. L. Cooke, Differential-Difference Equations (Academic, New York, 1963; Mir, Moscow, 1967).
B. G. Grebenshchikov, “The asymptotic behavior of the linear time-delay system solutions,” J. Comput. Syst. Sci. Int. 47, 187 (2008).
B. G. Grebenshchikov and S. I. Novikov, “On instability of a system with linear delay,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 1–10 (2010).
V. D. Furasov, Stability and Stabilization of Discrete Processes (Nauka, Moscow, 1982) [in Russian].
B. G. Grebenshchikov, “On existence of asymptotically periodic solution of a system with delay,” Izv. Ural. Univ., Mat. Mekh., No. 26 (5), 44–54 (2003).
F. R. Gantmakher, Matrix Theory (Nauka, Moscow, 1967; Chelsea, New York, 1960).
B. G. Grebenshchikov, “On asymptotic properties of a differential-difference system,” in Differential Equations with Partial Derivatives, Interschool Collection of Articles (St. Petersburg, 1992), pp. 57–67 [in Russian].
V. D. Furasov, Stability of Motion, Estimates and Stabilization (Nauka, Moscow, 1977) [in Russian].
B. G. Grebenshchikov, “On stability of non-stationary systems with big delay,” in Stability and Nonlinear Oscillations, Interschool Collection of Articles (Sverdlovsk, 1984), pp. 18–29 [in Russian].
A. V. Kim, W. H. Kwon, V. G. Pimenov, et al., Time-Delay System Toolbox (for Use with MATLAB), Beta Version (Seul Natl. Univ., Korea, Seul, 1998).
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Original Russian Text © B.G. Grebenshchikov, A.B. Lozhnikov, 2017, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2017, No. 2, pp. 3–15.
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Grebenshchikov, B.G., Lozhnikov, A.B. Stability and stabilization of one class of linear nonstationary constant-delay systems. J. Comput. Syst. Sci. Int. 56, 173–185 (2017). https://doi.org/10.1134/S1064230717020095
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DOI: https://doi.org/10.1134/S1064230717020095